Using "illegal" in the casual was that Josephine Fairley intends it (see below for more on this), the answer is:
In some European Countries: Yes, it is "illegal".
In all EU Countries: It is highly likely to be "illegal".
In France, Article L1132-1 of the Code du Travail ("the labour code") prohibits discrimination in employment (including recruitment) by reason of the prospective employee's physical appearance "de son apparence physique".
There are circumstances when an employer may justify discrimination on grounds of physical appearance, but that is also true of all the other grounds of prohibited discrimination, so it is "illegal" to discriminate on the grounds of physical appearance (including attractiveness) in France in the same sense that it is "illegal to discriminate on the grounds of gender".
For the rest of this answer I will ignore the possibility of justification. I am treating Josephine Fairley's claim as meaning "it will be at first sight unlawful..." subject to justification, rather than that it will always be unlawful, because that would not be true of discrimination on grounds of gender etc either.
Some systems of anti-discrimination law have a fixed list of "protected characteristics". French law is such a system, which happens to include "physical appearance". Belgium is similar, prohibiting discrimination on the grounds of physical characteristics.
For reference, a table of EU countries' list of such protected characteristics is found on pages 12-13 of Developing anti-Discrimination Law in Europe (European Commission, DG Justice, October 2013).
From that list it will be clear that some jurisdictions have an open-ended list. For example Germany which prohibits discrimination for a number of reasons including "... or any other ground". Such an open-ended list may be interpreted by the country's courts as prohibiting discrimination on the grounds of attractiveness. I am not able to read German, so I am not familiar with German law on this point - given that the question asks about law in Europe in general, I won't try to research it. The OP can always ask elsewhere if they wish to know more.
Europe in general
Directive 2000/78/EC ("the Employment Equality Framework Directive" sometimes known as just the "Framework Directive") requires EU Member States to prohibit discrimination on certain specific grounds (religion, belief, disability, age or sexual orientation, see Article 1). These grounds, along with race or ethnic origin (protected by Directive 2000/43/EC) and sex (protected, now, by Directive 2006/54) are a minimum level of protection required of all Member States, which is why France and Belgium are able to expressly protect additional characteristics.
All three directives prohibit indirect as well as direct discrimination, defined to mean a situation where an apparently neutral provision (i.e. on that is not directly discriminatory) puts someone at a "particular disadvantage" because, for example, of their sex, age or race. Indirect discrimination may also be justified in certain circumstances.
The problem for an employer of being "lookist" is that almost any criterion of attractiveness will inevitably put some people at a particular disadvantage because of one of the protected characteristics, particularly age and disability, but also race, sex etc.
For example, most (though not all) ways of assessing physical beauty will treat more young people as being "beautiful" than older people. I am not saying (as one commenter complains) that only young people are beautiful, but that an older person is more likely to be disadvantaged in a comparison of physical beauty. Similar remarks apply to disability (especially where the disabled person is obese) and race.
UK Solicitors Shoosmiths (riffing off a reference to Abercrombie & Fitch being investigated in France for their attractiveness policy) discuss just this problem from the point of view of British discrimination law. Their view is that it is very risky for an employer to try to discriminate on grounds of attractiveness.
An example of the kind of mess you can get yourself is the English Employment Tribunal victory of a girl discriminated against by Abercrombie & Fitch because she had a prosthetic arm. Note that this case does not set a precedent (it is a first tier case) and that it is about discrimination in employment rather than refusal to hire, but I think it makes the point neatly.
My own view, which probably does not carry much weight on Skeptics but is based on having quite a bit of experience in British discrimination law and having fought many cases, is that it will be rare for it to be safe to discriminate on the grounds of attractiveness without an "objective justification" such a justification would have to be proportionate.
On the meaning of the word "illegal"
Being pedantic, the term "illegal" is usually used by lawyers to refer to an action which has a criminal sanction. It is illegal to steal. Some jurisdictions do make discrimination illegal (for example the French penal code makes discriminating on grounds of physical characteristics in the supply of goods and services a criminal offence).
However, in most cases employment discrimination is not a crime. For example, in England and Wales, employment discrimination is a civil wrong (technically known as a "tort"). The discriminated party may be able to sue for damages for discrimination, but the discriminator does not face criminal prosecution. This is the reason for the scare quotes around the word "illegal".
A second layer of pedantry, which I have alluded to above but which has been pressed in comments, is that "it's illegal to discriminate on the grounds of gender" is not true in general. There are circumstances when gender (or as the UK legislation puts it "sex") discrimination in employment may be permitted, namely when the discriminator can prove that it is an occupational requirement and a proportionate means of achieving a legitimate aim.
In other words, if the employee can show (and in Great Britain there are rules on the reversal of the burden of proof at an early stage) that there has been discrimination, the employer must justify it. I am using "illegal" to mean essentially this: if you do this, you must be able to prove that you are justified (in the sense I have explained) or you will be committing a wrong (a tort or delict or however the particular system of law classifies it).
This isn't quite the same thing as walking out of a shop with items being "illegal" unless you bought them - that is a more complex situation involving the law of theft as well as rules on personal property ownership. I won't get into that here.
I think it is reasonable to test Josephine Fairley's claim using the same notion of "illegal" as she does.