Most of the previous answers make much harder assumptions than those actually found in nature.
I know of no reproductive process in the animal kingdom that doesn't have an egg or egg-analogue at some point.
Also, most of the cited situations (reptiles do this, mammals do that) have more exceptions than the answers imply.
While 'most' fish/amphibians/reptiles do lay eggs, many don't. While the platypus lays eggs, most mammals don't.
Some fish and reptiles (specifically some shark species and anacondas) hatch their eggs inside the mother, and then 'give birth' to live offspring. I don't see any significant difference (at least as far as this question is concerned) between that process and the process of a mammalian amniotic sack(egg-analogue) breaking, and then the mother 'giving birth'.
In the end, there is an 'egg', and it will 'hatch' and the offspring will come out of the mother whether still in the 'egg' or not. So the level of 'fish-like' qualities vs the 'mammal-like' qualities of the merfolk isn't really a factor. Instead, the answer is which fish and/or mammal is the mermaid the most similar too, in terms of reproductive biology. It could be mammalian, and still lay eggs, or fish/shark and still give live birth, or not.
Yes, it's entirely plausible for a species of merfolk to have relatively softshelled eggs, if that's how you want them for the story. And it can be adjusted anywhere along the entire range of mammal, amphibian, reptile, or fish spectrum as needed.