How to change the colour of a png image in any adobe software

What you're experiencing is the most easily perceived difference between vector and raster images. While the former describes shapes in an abstract way, the latter deal with images as a collection of rectangular dots, called pixels.

You can surely export the images in a larger resolution which raises the quality at the expense of file size. But if you want to retain the quality of the output then you shouldn't use JPEG or PNG but rather a file format capable of representing vector graphics. If it's for printing then PDF is usually a choice most companies can work with. Besides, it's the native format for Illustrator anyway.

Side note: You don't really want to export vector graphics in a file format like JPEG anyway because JPEG is primarily intended for photographic images. A direct consequence is that JPEG performs extraordinarily bad with line drawings, sharp edges and high contrast which is not uncommon with vector graphics. PNG is a much better choice here if you need a raster format and usually even compresses such images better.

Use a Hue/Saturation Layer Mask and turn the Lightness up and Colorize "ON". That will allow you to change the colour of a black object.