I don't entirely agree with the concept that you have to use exclusively on style of corner throughout your app. I think it depends on what you're trying to accomplish with your design.
For example, rounded corners are very popular on input boxes and site-level wrappers. They lend a nice, smooth appearance that is appealing to users and has a sleek, modern feel. However, that same site could have a layout of card structures linking to various parts of the site. Coupled with a subtle box shadow, this can be a nice analog that is very intuitive to users with the bonus of being visually appealing.
The bottom line boils down to opinion. Rarely do corner styles play a large role in usability of a site which, in my humble opinion, should always take priority when designing an application, be it web or native. Do what makes the most sense to you as long as you're sure it won't negatively impact your user's experience.