Keep in mind that the first amendment does not apply in your situation. Like almost 100% of the time the 1st is used as an argument by people, it is being misapplied by your tenant.
The first amendment is applicable ONLY to how the government may restrict what you say. Since you are not a government agent, the argument is irrelevant. The first amendment doesn't give you the right to say what you want wherever you want, that's a gross inaccuracy. It's only applicable when the government is involved.
But since your tenant is already being evicted for non-payment of rent, it's probably easier just to temporarily remove the board or restrict postings to just your own until she moves out. No sense in making life more difficult for everyone, especially you.