You are correct that it is a good idea to use teflon tape on the threaded fittings, however no sealant is required or advised on the quick-connect part itself. These are designed to seal using internal o-rings.
There are multiple types of quick-connect air line fittings that at first glance appear similar, but are not necessarily compatible. It is very likely that you have connected two incompatible types, resulting in a leak. This is very common.
The two "main" types are often referred to as "automotive" and "industrial", though each can have sub-types that are typically letter designations, e.g. Industrial can be M, H or G. There's also a type called ARO, and one called Lincoln, but these are somewhat less common (at least in the US consumer market).
![air fittings](https://i.stack.imgur.com/QDCNP.png)
(image from Legacy Manufacturing)
The documentation for your air compressor may specify exactly which type of fitting you have. Otherwise head to the local big box store with your male coupling that leaks, and look for a similar-while-slightly-different male coupler to try.
Legacy manufacturing has published a very informative guide on the evolution of the various types of quick-connect fittings that lists the common types in use, with some images of each.
Milton Industries has also published a document that may help identify the type of plug you have, though their letter designations may not exactly match those of other manufacturers.
Best Answer
If the metal button that you refer to is this one:
Best guess is that you press this when engaging the end of the gun onto and when disengaging the gun from the valve stem. When the gun is fully pressed onto the valve stem letting go of this button helps clamp the tip of the gun onto the threaded end of the valve stem.
Again this is an educated guess because I've not seen your gun first hand. You could probably look at the tip of the gun assembly and toggle the metal tab back and forth and see right away what it does.
Edit
In looking more closely at your picture it appears that the tip of the gun can have various ends slid into it and this lever is designed to hold the tip in place. Pressing the tab (button) releases the catch and allows removal of the tip.
When I refer to the tips I mean one of these from the kit: