Can a False Dilemma have if this then that language as well?

Mo me, the statements have so many places of flawed thinking, it is difficult to pick one or two. Certainly, the false dilemma (fallacy of the false alternative) is something to consider AFTER getting the agreement of participants to the definitions of the operative terms. I note the undercurrent of both is the premise the government should tell private businesses when to close, order their customers not to go to a business, or force an allegedly free population to wear a mask. This is in addition to the whole debate about whether masks "work" in this instance and would require discussion.  

Yes, a false dilemma can have "if" statements because of what is implied. The meme you referenced implies one of two options: "work" and "don't work." Consider the following:

If masks work, we should encourage people to wear them.

If masks don't work, we shouldn't encourage people to wear them.

This is very different from the meme. In my above example, I have created a dilemma, but not a false one because of the implied definition of "work." "Work," is a threshold term where it is a legitimate binary: works or doesn't work. Admittedly, it is a poor choice of the term but not fallacious. Here "work" is synonymous with "has a positive effect." Also, there are no conditions in which they work, so it is poor communication, but still not a false dilemma. In the meme, "work" is synonymous with "prevents 100% of COVID," and of course, this leaves out the 0-99.99...% options.

These kind of sophomoric memes are best countered with simple analogies showing the flaw in the logic since people who agree and share these memes wouldn't even begin to understand what I wrote above. Condescending? Perhaps, but also unfortunately true. My response would be:

If seat belts work, why do people who wear them die in car crashes?
If they don't work, why are we forced to wear them?

If you are more interested in making the person look stupid than having them agree with you and see their error in reasoning, you can add:

I'll give you more than a minute. I suspect you'll need it.