An "animal companion" is a special ability that mostly druids and rangers have. It is irrelevant to this discussion as your rogue does not have it. See What feats can I use to get a companion of any sort? for a list of abilities that actually grant companions in the technical sense; these guys stick with you and help you because they're your nature buddy. The rogue, however, is basically in the same boat as someone who wanders down to the local zoo and convinces them to sell him a tiger. He has no specific control over it and it has no specific reason to care about him.
A PC can purchase and try to keep around any other kind of creature that's available for sale and they can afford, from animals to human slaves. Note that things are not available because "they are in a book the player has" - you're the GM, put on the big boy pants, you control the game world and what is available and how in the markets of your game world. If such a creature is available by sale or by capture, whether you can use them effectively and whether they kill you in your sleep or not has a lot to do with other rules systems and roleplaying.
If you have an animal (a monster of type "Animal" - not magical beasts, not anything else) you can use the Handle Animal skill to teach them specific tricks over a long period of time and to try to command them in combat. There are a couple magical beasts that have an exception stated to this (hippogriffs, familiars) that can have Handle Animal used on them in specific circumstances.
If you have anything else, then it's pretty much you convincing them through other means to help you out. This is one part GM fiat, one part bluff/intimidate/etc (for intelligent creatures). Charm or dominate, or the Leadership feat, or other specific rules in the game that allow for influencing other creatures apply as normal.
This is one of those things on which D&D bows to the DM's authority over their world. D&D has provided the creature, but not all the details of its life cycle, ecology, mating habits, regional variations, history, origin, etc. ad nauseum. Different editions have given more or less detail on these aspects of creatures, and this is the level of detail 5e has settled on.
So, in a phrase, "ask your DM," since it will vary according to what the DM wants for their setting. In one setting wyverns might be like reptilian mayflies who live only a year and die every winter, while in another setting they can live thousands of years, and in a third setting they have a lifespan similar to cats, etc.
Best Answer
You can definitely have a hippogriff as a companion--there's even an archetype for it.
But the text is silent on hippogriff egg incubation periods, so it's up to you as GM to decide.
If you're more of a storytelling GM, decide based on what's convenient timing for your game—otherwise known as "it incubates at the speed of plot."
If you're more of a simulation-minded GM, you still have to make it up, but you should find a number that you can accept as sensible. Perhaps research the incubation periods of very large and very small birds to find a ratio or power relation, then scale that up by your estimate of the mass of a hippogriff egg.
Either way, it's completely up to you—they're imaginary creatures, and the game doesn't care enough to say, so there is no higher authority than the GM on this question.