What does a flashing button mean on the Blackmagic ATEM 4 M/E front panel?
tl;dr
It indicates that ME 3 PGM is selected since there is no buttons for M/E 3 and 4.
Blackmagic ATEM Front Panel Tutorial
The switcher's front panel doesn't have a button for all routable inputs/outputs. If you click an AUX in the menu of the ATEM Software Control, you can see that you have 60 items in the dropdown, while there are only 36 buttons on the front panel.
Media Players and M/Es
You have 4 media players and 4 M/Es but there are only buttons for the first 2. For selecting the other half you have to use the same buttons by double pressing them. So to select Media Player 3 you have to double press MP 1. Buttons that work this way:

MP 1 -&gt; Media Player 3
MP 1 KEY -&gt; Media Player 3 Key
MP 2 -&gt; Media Player 4
MP 2 KEY -&gt; Media Player 4 Key
ME 1 PVW -&gt; ME 3 PVW
ME 1 PGM -&gt; ME 3 PGM
ME 2 PVW -&gt; ME 4 PVW
ME 2 PGM -&gt; ME 4 PGM

The mixer is really nitpicky when it comes to registering double presses. If you press the button to slow or too fast it won't recognize it. You have to press them in a specific rhythm with around 350 ms delay between the two presses. This might sound weird but it is true. I had to measure this since it seemed so unreliable. 180 ms doesn't work, 520 ms doesn't work. Between 250-450 ms you are good. Thanks to this neediness I would not recommend the front panel for AUX routing for these inputs when the timing is critical.
Key Masks
Every key's mask can be routed to an AUX. We have 4 upstream keys for the 4 M/Es and the 2 downstream keys on M/E 1. This means (4*4)+2=18 masks. All of these can be selected via the KEY MASK button. This button always flashes if any of the aforementioned key masks are routed to that AUX. On each KEY MASK button press it will cycle through the key masks, selecting the next key according to the order in the menu dropdown:

If the previously routed signal on that AUX was a key mask, the button press will go to the next one (below it). If it reached the end or the previous signal was not a key mask it will start from the top.