It has happened to all of us as photographers - professionals, amateurs, and parents. We ask a child smile. We ask a kid to just say cheese. We get back the pained uplifting of the corners of the mouth and the dead eyed stare that says "take the picture already." So we take the picture and pretend that faux smile or teen indifference is the "look" as if every mother doesn't want to remember most the squinting eyes and broad grin of truly happy moment.
So how do we get there? Before engaging kids with a camera, I used to do it with a deck of cards and other magic tricks and it always took time to warm up the audience. Some times, I'd do a little back and forth with names. Some times, I just gave them permission to make noise. I always started with a trick that they thought they could figure out. What I've learned from those years, is if you want the best results with child portraits, you need to warm them up, give them permission to laugh and make noise, ask them about their likes, dislikes, etc.
Sure, if you actually get a genuine smile, their eyes can get a bit squinty. It is how you know that it is a genuine smile. Once you start getting that smile - not the selfie smirk or indifferent line - you can work on getting their eyeballs back. Take the picture as the smile begins to recede since the eyes will open up before the grin goes away.
So take it from a guy who can and will pull a coin from behind a child's ear if asked, sort out the lighting (preferably before bringing in your subject), get the smile, then get the picture. Clicking away while they are self conscious and then asking them not to be self conscious is not a great strategy. So ask them who their favorite teacher is and why, ask they who their least favorite teacher is and why, ask them is their is anyone cute in their grade, ask them about themselves and their lives as though you find interesting ... and better yet, find them interesting. You will get better pictures of them and you will make their mothers happy.