This appears to be somewhat true. Death rates increase when you lose your job, but 40,000 people per year is an order of magnitude higher than my estimate.
This Quora question answers an identical question and concludes that a 1% increase in unemployment results in about 1,500 excess deaths per year in America. My answer follows the same logic as the Quora answer. I have checked his sources, but I am forced to find my own data for mortality rates because his link is broken.
There are roughly 162 million workers in the US, therefore a 1% increase in unemployment corresponds to 1.62 million workers losing their jobs. According to this CDC data, for every 100,000 people aged 25-64 roughly 400 of them will die in a given year. That number comes from averaging the mortality rates for the age groups I assume make up most of the labor force. Therefore, for a given sample of 1.62 million working age people, we expect 6400 of them to die in a given year. This meta-analysis indicates that your risk of death increases by 63% when you lose your job. This means that 10,000 people will die instead of 6400, an increase of 4000 Americans per year. In order for this claim to be true, the increased death rate would have to be 630% instead of 63%.
The Quora answerer looked at mostly the same data I did, and his answer is of the same order of magnitude as mine. We both make some simplifying assumptions to get to an answer, and I believe his numbers about as much as mine. Take both with a grain of salt.
As a further warning to take this logic with a grain of salt, the meta analysis has a pretty detailed discussion of confounding factors.
Many researchers continue to argue that the unemployment-mortality association is spurious. These scholars argue that health selection into unemployment operates through health behavior variables rather than in a direct manner (i.e. the “latent sickness hypothesis”) (Jusot et al., 2008).
In more plain English, people with unhealthy behaviors may be more likely to lose their jobs; Alcoholism can get you fired and lead to an early grave, but getting fired wasn't what killed you.
A third reason to take this with a grain of salt is that when unemployment increases, underemployment increases as well. This logic may exclude some number of people who die after they take two crappy part time jobs after they lose their one good job. Those people are not counted in the U3 unemployment rate. If your child or spouse dies after you lose your job and your health insurance, that death would also not be counted.
Summary: A 1% increase in unemployment results in a few thousand excess deaths. I am uncertain how many thousand exactly.