Do you know of another company that deducts income taxes like this?

I have not heard of this being applied to business meetings, but if your your company is based in Massachusetts and you are a sales rep, for example, and you have clients in Connecticut or Rhode Island and you travel to those clients, then the income earned in those states is taxable to those states. You would get a credit for takes paid to those jurisdictions on your MA return.

The biggest employers that I have seen this on are construction contractors that have jobs in many states.

Massachusetts, as well as other states I'm sure, have a form to allocate income based on the number of days spent in the state.

That's the most ridiculous crap I have ever heard. You are not getting paid by an outside company while outside of NY, so the taxes should stay at a NY rate. You haven't earned money from the Massachusetts business meeting, you weren't paid by some Massachusetts corporation for participating in the meeting, you were just attending a meeting! I think your accountant is way off. I would call your own personal accountant, the one that does your taxes, and ask them about it. I have never heard of such a thing in my life and think it sounds very suspicious.

That's weird and makes no sense. No, it's NOT how they are supposed to be doing it.