There are a few things to consider for button designs, but I don't know what you know or don't know so I'll list some general stuff here:
Scale - you need to know what he's using the buttons for, if it's for ios then you need to design them 3x bigger as retina screen has a higher pixel density.
File format - In general it should be in PNG, or SVG
Size - all buttons should be the same size in terms of pixel, i.e. all buttons should be 100x100 then you fit your icon/text inside it. This way if the button design changes then it's still the same size
Adobe XD + Zeplin - seeing as you have Illustrator I'm going to assume you have the Adobe Suite. You should be designing assets in photoshop/ Illustrator then lay them out on Adobe XD. If you install Zeplin as well it will allow you to easily manage your assets and export them correctly. If you tell Zeplin your app is in ios then it will automatically upscale x3 for you. Most UX designers use Sketch but Adobe XD has improved massively within the last year and is definitely worth using if you don't want to pay extra for it. Even if you don't have Adobe Suite, Adobe XD is free to download so there is absolutely no reason why any UX should be done in Illustrator