Chronograph pushers are typically spring loaded with a coil between the button and the case. But what keeps it in place? In most cases, all that is keeping your chronograph button from jumping out of the case is a tiny little part called an E-Clip (also called a C-clip, and technically called a Circlip).
“Circlips (also known as retaining rings) are types of fastener which look like flexible, open-ended rings. Circlips are typically made from stainless steel and similar metals. Alternative names for circlips include retaining rings, retaining clips, C-type, Jesus clips, snap rings, and E-type.” –source
The pusher stem typically has a groove for the C-clip to be positioned. As you compress or press in the button, the C-clip rides along with the pusher stem, and when you release the pusher, the spring pushes is back out, but prevents the button from coming out of the case.
Below is a macro close-up example of the C/E clip. Notice on this example there is a flat part in the middle. Some of the clips have a longer center prong, which makes it look more like an E than a C. This also makes it easier to hold with your tweezers, helping with enough leverage to squeeze it on to the pusher stem without it slipping out of your tool. Although, that will still happen! Make sure you have another spares because they are hard to find when they fly away, and don’t beat yourself up about it!
Pro Tip: If you happen to loose one and need another in a pinch, you can usually borrow one from another chronograph watch until you get your supply of clamps replenished.