I assume that, if the White Eno eccentric hub moves the hub center in a circle around the axle, then it could potentially position the wheel higher or lower rather than just forward or backward. This would require the brake shoes to be adjusted each time the chain is re-tensioned. Also the attitude of the bike changes. The rear fender could be out of position. Am I missing something? Is this a real problem?

Anyone know of another solution for making a fixed gear setup on a bike with vertical dropouts?

your assumptions are correct. the Eno hub axle defines a circle around the attachment points and the consequences are correct as stated.

i don't have a problem with my two bikes that use it, but i don't run fenders or a rear brake. alternatives involve a dedicated chain tensioner or eccentric BB or compromises concerning gear ratios ("magic gear", see Sheldon B.).

BTW, the amount and frequency of chain tensioning necessary to effectively MAINTAIN an existing gear ratio is negligible. i find most if not all the fuss involving the ENO hub is when deciding upon the proper chainring/cog/chainlength issues, which, if you are like me, is not very often. once set up with the ratio desired, the wear on a chain or cog or chainring that demands hub re-positioning has little real-world effect on "attitude" changes or brake positioning (which is trivial).

and i can't say whether or not you are "missing something", but i can't find my 3/8 socket wrench. you don't, by any chance, have it, do you?

While it will make a difference setting up fenders and brake shoes the first time, it's not the problem you envision because the chain length doesn't change much over time so the rear wheel will be repositioned in the same place each time you take it off and put it back.

If you're lucky and the cam is near the top or bottom, whatever difference there is will be mostly horizontal and not up and down. In fact you only see serious vertical changes at the extreme forward and back positions, which it should be possible to avoid with a half link, or a chainring 1 tooth larger (if it comes to that.

And ... having a half link in the chain will be your friend..

I forgot all about the half link...brilliant. Sounds like this issue is no big deal. Thanks all.

Even without the half link it wasn't an issue. The rear wheel is returned to the same position, or very close every time, so nothing moves once it's set up right the first time.