The "rimless" 2-piece brass version of US Army Engineers (like yours) was made during the late-1800s through early-1900s. A rimless one which is extremely similar to yours (but not quite exact, due to tailfeathers) is shown in the Alphaeus H. Albert button-book as button EG-9, with a late-1800s Waterbury Button Company backmark.
I checked the Waterbury Button Company's catalog from the year 1900, and the US Engineers button in it has a rim... so the WBC had switched to making the rimmed version by that year.
Your button's "Extra Quality" backmark is written in serifed lettering (not plain-block lettering), which is an additional indication that it is from no later than the very-early 1900s.