The answer is that yes, they can require you, but you are not being unreasonable to make a fuss about it -- like you, I don't like giving anyone direct access to my bank account, as I know from experience that it can be difficult to get them to turn it off or straighten out billing errors.
What is the reason that they are requiring ACH? Sounds like it's a condo association, which may make more sense. FWIW, I am on the board of a small condo association. We used to require ACH. Then last year we stopped requiring it -- and all of a sudden, we started to have a bunch of issues with people "forgetting" to pay on time. That made things really difficult; I mean, our bills need to get paid whether residents are paying on time or not, you know? And we are small and trying to keep fees low, so we don't have a huge slush fund in the bank account to tide us over (in fact, we're trying to build our reserves back up). So this year we went back to requiring ACH.
My own approach is that if the relationship is ongoing/permanent/etc., I am ok with ACH -- I signed up for the condo ACH, for ex., because I have a condo there that I am not just going to walk away from one way or the other.* OTOH, if it is a relationship I may want to terminate at any time (karate, gym, etc.), then no way.
*Note that I am also on the board, which helps me make sure things are being handled appropriately, i.e., "trust but verify."