What alternatives do exists for properly squeezing lemons

You can try a few things.

1. Use just the zest from the peel, and then slice and remove the pith and use just the inside.
2. Pre-boil the lemon peels, then add them to your main dish.
3. Remove the lemons before the bitterness gets too strong (i.e. when you tasted the dish the first time) but while there's still a strong taste.

This works because the citrus oils (which are a major primary flavor contributor) are just in the top layer of the peels, while the acid is in the inside. The pith is just plain bitter.

Lemons are quite sour, while Meyer lemons are much sweeter and less acidic. If you substitute directly, it'll have a dramatic effect.

For example, suppose you start out with a dessert made with lemons that has enough sugar added (or little enough lemon juice) to make it the right sweet/sour balance for you. If you replace the lemon with Meyer lemon, it'll be way more toward the sweet side, and won't have the sourness to counteract it anymore. So unless you want that big change, you'd probably want to reduce the sugar and/or increase the Meyer lemon; exactly how you decide to do that would depend on the dessert.

If you substitute in the other direction, Meyer lemon to lemon, you'd likely have to reduce the lemon and/or add a bunch of sugar to keep it from tasting more sour.

Either direction, I don't think it really seems like a good direct substitution. It's a big change, so whether it's a good idea totally depends on whether the original recipe suited your preferences and whether you're okay with a broad range of sweet/sour or have something more specific in mind.

The only things where a direct substitution seems like a good idea to me are things where you're only using the zest. At that point citrus is pretty much all interchangeable, just a different flavor.