One could make a First Amendment challenge to mask requirements through either the Free Exercise Clause or the Free Speech Clause. Neither approach is likely to succeed.
Because going without a mask is not recognized as "expressive conduct," it is not protected by the Free Speech Clause
A free-speech challenge would likely also fail for two reasons. As you correctly suggested, the First Amendment protects more than just speech, also protecting "expressive conduct," such as flag burning, dancing, and wearing armbands. Of course, literally any conduct could have some secret expressive meaning in the mind of the person carrying out -- "I shot him in the face to say I didn't like him" -- so we have a question of where to draw the line between what expressive conduct does and does not receive the strong protection the First Amendment affords to speech.
The Supreme Court detailed that test in Texas v. Johnson, 491 U.S. 397 (1989), and it's now generally formulated as asking two questions:
- Did the speaker actually intend to convey a particularized message through his conduct?
- Are the people who see the conduct likely to understand that message?
At the Sturgis rally, or at the statehouse protests over COVID restrictions, or some similar event that is explicitly opposed to masks, refusing to wear a mask goes a long ways in communicating an opposition to the mask requirements.
But in the vast majority of cases, no one you run into in the normal course of daily life is likely to recognize that you are not wearing a mask because you are trying to communicate a message, let alone decipher what that message is. Do you believe that mask mandates are tyranny? That COVID-19 is a hoax? That life is meaningless and we should all welcome the hastening of human extinction?
None of that is clear to the average viewer, which is who the courts are going to be concerned with. Because refusing to wear a mask is generally insufficient to convey a specific message, I'd argue that it is not expressive conduct.
Because mask requirements are neutral as to religion and generally applicable, they do not violate the Free Exercise Clause.
A religious challenge ("My religion prohibits wearing masks") is likely to fail because "the right of free exercise does not relieve an individual of the obligation to comply with a 'valid and neutral law of general applicability on the ground that the law proscribes (or prescribes) conduct that his religion prescribes (or proscribes).'" Employment Div. v. Smith, 494 U.S. 872, 879 (1990).
This means that if the mask ban generally applies to everyone and you just happen to belong to a religion that forbids mask-wearing, you can't use that affiliation to escape the law's requirements. (The outcome may be different when you run the problem through the Religious Freedom Restoration Act or state-level analogues, which impose more stringent tests for infringements on religious liberty.)