The question you will ask has a lot of effect on what type of scale you use. Most of my answer comes from research into surveys, so for instance when you use a questionnaire to know the market impact of a certain product. (When you are walking in a store you sometimes have people asking questions about a certain 'soon to be released' product.)
When to use a 5 point scale (Likert scale)
A 5 or 7 point scale is best used when you want to measure latent constructs. In short, latent constructs are questions about: Opinions / feelings etc. These are unobservable individual characteristics.
When to use Yes/No questions
The Likert scale is best used when you want to ask latent constructs, so a yes/no question is best used when you can confirm an answer. 'Are you married' 'Do you have kids'.
What should you use now
Given that you want to ask if people are satisfied with the application, the best course of action will be a Likert scale. Finally, if you want to check the reliability of your 5 / 7 point Likert scale you can use Cronbach's alpha
I could not find any 'research' specifically on applications, however I did find something regarding the 5/7 point Likert scale that you can use to convince your stakeholders.
In current practice, most rating scales, including Likert-type scales
and other attitude and opinion measures, contain either five or seven
response categories (Bearden, Netmeyer, & Mobley, 1993; Peter, 1979;
Shaw & Wright, 1967).
My personal opinion is that you should preferably not use a popup to ask this question, having a UX test with the target group in a controlled environment will give better answers then a (usually annoying) popup. If you do want to implement the question I suggest you do it at the end of the funnel, so after they used it to do a certain task and not at the startup.