I posted the same type question here and in Plumbing last year when planning a bath remodel. I found this drain and have had it installed; the bath is almost complete so I cannot vouch that it will work as I intended. I got it because the drain cover is easy to remove to clean out hair. don't know how easy it would be to replace your drain with one of these and just clean it out yourselves periodically. My thinking was I could put a mesh screen under the cover to catch all the hair, but haven't tried that yet. My plumber and tile setter loved the drain by the way.

Here is a link that might be useful: Ebbe drain

We bought a "strainer" from Lowes that the drain cover snaps onto and then you put it back in place. It was around $3. It catches all the hair and gunk and we pop it off and clean it out about once a week. My daughter and I both have long hair so ours was constantly clogging up too. Ours is a shower so it is a little bigger than a regular tub drain. I am not sure if they have square ones but the metal is easily bent so you could probably make it work. and it is only about 3 bucks so it is worth a try.