The article you linked says that its source is an article on Spinzon which says that its source is an article on TheRightists.com from January 6th. In fact, they have mostly the same text.
TheRightists.com describes itself as a "HYBRID site of news and satire", saying
part of our stories already happens, part, not yet. NOT all of our stories are true!
That doesn't prove any part of the article to be false, but it certainly means you can't take any statements in it for granted. In particular, the article quotes extensively from a supposed spokesperson for Refuse Racism talking to the New York Times. The claims in the YourNewsWire article's headline and first few paragraphs about Refuse Racism's goals seem to be referring to these statements. As far as I can determine, these statements do not appear in any New York Times article. (I found no results for text searches of recent NYT articles with key phrases, and Snopes says the text of those statements "was not published in an article by the New York Times".)
If you read the articles more carefully, they don't quite say that Rosie O'Donnell, Debra Messing, Ed Asner, and Michael Shannon "have threatened to go on strike until Donald Trump resigns the Presidency". Instead, they say that those people
have attached their names to an effort calling for a month-long protest to stop President-elect Donald Trump
and that the calls "for an all-out strike that would span the entire movie industry" are instead from "other Hollywood celebrities" (emphasis mine). Unfortunately, they never mention who these other celebrities might be.
This slightly different claim about what the four named celebrities have expressed support for (signing support for a protest, not necessarily a strike, nor by all of Hollywood, and one month, instead of until Trump resigns) appears to be true. The Refuse Racism website highlights a number of signatories of their call to action, and those four people are listed among them.
This call to action does mention "walkouts from schools and work" as a possible action for people to take, one of many suggestions.