Can they "demand" it? Sure. Can they force you to comply? In what sense? They certainly can't send a SWAT team to break down the door of your house, tie you to the chair, and turn the camera on.
Conceivably they could fire you. That seems to me to be a rather extreme response to something like this, but I suppose if you refused and the boss insisted and it turned into an escalating argument, that could happen.
You say that you are in the US, so your contract is probably "at will", which means the company can fire you for any reason not specifically prohibited by law. "Illegal" here would mean things like racial discrimination or sexual harassment, which I doubt apply here. (Unless the boss is demanding that you turn on the camera and then stand in front of it naked or something crazy like that.)
Really, I'd think if the boss says, "hey, turn on your camera" and you said, "Oh, I can't, I'm wearing my pajamas" or "this room is a mess, I don't want anyone to see it", that everyone would just laugh and move on. If you have some good reason, say what it is. If it's something you can fix for the next conference call, fix it so this doesn't become an ongoing issue.
I work from home. My company occasionally has video calls. The first couple of times we all turned on the cameras, but after that the wonder and excitement of seeing each other quickly wore off and now everyone leaves them off and we just show screen shots.
(When I first got this job, a friend asked me if I sat at the computer all day in my underwear. And I cried, "Oh man, now you've put an image of all the other employees sitting at their computers in their underwear when we have conference calls. If I worked with a group of beautiful young girls that might be a pleasant image. But I work with a bunch of overweight, balding old men.")