On the whole we tend to avoid them even when we're staying onsite, but there have been exceptions. There's certainly an argument for attending the morning ones during the first few days when you'll be awake at the crack of dawn, but only if you think you'll be disciplined enough to hop to another park once the crowds start to build up. For me, it's too bitty. I don't mind visiting a park during the morning and another for the evening after a break, but I'm sure I wouldn't feel ready to move on mid-morning.
We attended an evening EMH at AK because it gave us a rare opportunity to tour after dark. An hour-long storm early on emptied the park and our patience meant we had it virtually to ourselves. It was magical and we got some fantastic photos. Parts of the park were closed off, though, and that's one of the downsides to EMH's in general. Are they offering them at AK at the moment? Not sure that they are, which is a real shame.
We've attended both evening and morning EMHs at MK and Epcot. The morning EMH at MK because we had an ADR at Crystal Palace for our daughter's birthday, the evening one because we had a one-night stay at GF for mine and thought we'd take advantage. The Epcot forays were both against our will when we were visiting with friends. The only one I enjoyed was the MK evening one and that was actually an E-Ride Night (in the days before evening EMHs when numbers were limited to 5,000). I would consider one of the 3am MK ones, but I'd arrive late and stay until the bitter end. If you think you might be flagging by 11pm, you'll just experience a packed park and be leaving with the hordes. The other times, the parks were mobbed. I've never seen crowds like it. We staked out our place for IllumiNations very early, so had a spot next to the railing, but the crowds were 10 deep behind us.