Is there a free egg in the launch login rewards?

The egg is not handed to you on login. It arrives in your in-game mail, and the rare Prismana egg only unlocks once you complete the daily check-in tasks.

So two habits matter in your first days. Open the mailbox, where the launch mail carries a Sparkling Egg. Then do the daily check-in tasks through to the end, because skipping days is what keeps the Prismana egg locked. The egg is limited, so treat it as a launch window reward rather than something you can come back for later.

Other free eggs once the login egg is claimed

The login egg is the first of several you can earn without paying. Sparkelf is given to every player through the Sparkelf event, as an egg awarded at 200 personal Spark points. Reaching Homeland level 15 rewards a mysterious Prismana egg. The Prismana Pact launch selector opens at account level 40 with the Junior Wayfarer Pathfinder rank (also called Junior Excursionist) and lets you choose one free rare Prismana Aniimo between levels 40 and 45. The Dazmand event rewards a perfect Dazmand egg.

Operation Egg Heist is where eggs are farmed rather than collected once. Teams extract special eggs, including rare Prismana and Umbral (Darkler) eggs, and they are awarded on extraction. Normal difficulty rewards can include an Umbral egg, and there are three Prismana egg rewards to work toward.

The exclusive Prismana egg at level 10 of the Advanced Companion Handbook is not one of these. That handbook is the premium track.

Not settled: The launch egg is described both as a Sparkling Egg sent by mail and as a rare Prismana egg from daily check-in tasks, and whether these are one reward or two separate eggs is not settled. The Prismana Pact selector is also unsettled: it offers either 14 or six Prismana options.