If it was requested I go on a weekend trip with my boss & the members of the Board, can I say no? If this is something that is assigned to me, am I risking my job by telling them no?
I was going to go, but I cannot find child care for the weekend so I dont see how this will work. My job description says "...and any other duties as assigned". So if they assign this, am I stuck?

Yes, you can be required to go on this trip. If you don't, you can be disciplined up to & including being terminated for not doing so. (unless you would have a binding employment contract to the contrary)

Your employer's concern is the business & not your child care problem.

Betty is correct, however, that doesn't mean you can't ask your boss for an exemption from this trip because of personal needs. None of us can know what your work environment is like or how well that request will be recieved.

They have the legal right to discipline you over it, including termination, under at-will employment. Conversely, you have the right to quit.