Well this is kind of the "gotcha" here. Typically when I make a case for proper QA it fighting to demonstrated dedicating XX hours to QA per sprint. Usually you can put together the math based on other's research to figure out what is going to be a pretty good balance between time invested and bugs squashed.
Making this time a person
Most of the time those hours can be committed by engineers and actually work well enough. A dedicated QA person is more efficient, but if your need isn't enough to justify it then engineers time will do.
For you to sell your boss on a dedicated QA person you really need to demonstrate you need enough QA time to cover a fulltime position. Outsourcing a part time QA time is a REALLY hard sell, and in my experience one that typically hurts more than helps.
Making the most of QA time
So you have your entire dev team drop what they are doing and do manual QA... This... is bad. If you can dedicate a bunch of time to manually test everything you should be able to commit time to making proper unit and integration tests! Usually the cut best bang for your buck is 15% to 20% of your dev time dedicated to cleaning up old code, bug fixes, and non-production projects such as building out tests, database clean up, optimization, etc. (The crap that needs doing that doesn't directly make money)
As a developer good TDD can be invaluable and save you HOURS of debugging issues. You can reliably count on each part of your code doing it's job, and in the event something breaks you have a test that says what broke and where it broke, even before you spin up the UI to see that it broke. This takes your necessary QA time WAY down. Now instead of "testing all the things" you are really just making sure the UI is in good order... (Which actually works against you getting a QA person, unless you just dump building out automated testing on this person... which isn't ideal, but WAY better then no automation)
A dedicated QA person is a hard sale and you need to show your boss all the engineer's time dedicated to testing would go further in the hands of a QA person...
But that's really not your problem... You need to automate your testing to just reduce your QA time entirely.