Well, there is a way to both select and fill at the same time yet it may not be everyone's cup of tea or for the faint of heart.
This can be achieved by using the capability under File > Scripts > Script Event Manager along with a very simple Action
1) Create an Action that all it does is a Fill command using Foreground using Cmd +Opt keyboard shortcut (Cntrl + Alt for PCs) Remember where you park your Action 
2) File > Scripts > Script Event Manager
--- Click on the checkbox in upper left corner labeled "Enable Events to Run Scripts/Actions"
--- Click on the dropdown box "Photoshop Event" and select the option "Add an Event"
---- Fill out the dialog box. The upper box is just a descriptor name (you can name it anything) the second box must be set to lower case "setd" without the quotes as shown. This is the event for a simple selection (not add to, subtract from, exclusive or select)
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------Click OK
------ Back in the original dialog box click the check box to run an Action and select the Action Set and Action you created in #1 above. I filled in where mine was located
------ CRITICAL CRITICAL CRITICAL after doing these steps you need to click on the "Add" button to have this event activated
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3) I will assume you have already set the grid and a Layer above for the fill action you need
4) Select the Magic Wand Tool with a simple select mode, contiguous, and sample all Layers. I used "Point Sample" and tolerance of 8 yet this depends on the background color of your grid. Setting Anti-Alias is optional
5) Select desired foreground color from the foreground color chip in the toolbox
6) Click in the middle of the grid area you want filled and the grid square will be automatically filled with the foreground color. Keep picking square to fill. You can change the foreground color and click grid cells with the new color
7) When done, go back into Files >Scripts >Scripts Event Manager and turn off the event by unclicking the checkbox in upper left corner (you will regret it if you don't do this last step)
Here is a simple grid that I filled one click at a time and changing foreground color as I went along:
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NOTE: This event is also triggered by some other actions such as deleting a selection so the downside of this approach is the overhead of turning it on and off and not doing other Photoshop work in the meantime. I believe the Event Handling could also be turned on and off with a simple Action if that is helpful.
Just another way to consider and hope this helps
John Wheeler
