Sun's Highest Point on June 21
Well, kind of, depending on what you really mean.

We're being rather vague with our terms, and so it shouldn't be surprising if the answer doesn't meet our expectations.

We could interpret all these terms to conclude that such a situation will only rarely, or even never, happen.
But allowing some reasonable latitude (pardon the pun) in our definitions, we can come up with a reasonable response of "yes". 

The Sun can only be "directly overhead" of one spot at a time, where the line between the Earth's center and the sun intersects the Earth's surface. Because the Earth is spinning on its axis and also moving in its orbit,  that spot is constantly changing latitude and longitude. 

The Tropic of Cancer isn't a single-point "location" but a circle that stretches around the Earth. 

At the exact moment of the summer solstice, the Sun is directly overhead one point on the Tropic of Cancer and one point only. That point slowly spirals down through the tropics until it reaches a corresponding point on the Tropic of Capricorn at the winter solstice. And then it begins the trip back up.

That's what happens in the geometrically-idealized definition of "directly overhead".  We can allow a small tolerance, and as the tolerance gets bigger, the spot of "direct overheadness" spreads out into a disk, which covers more and more of the Tropic of Cancer around the moment of the Solstice, due to the spot's E-W motion being much greater than its N-S motion.

Although the spot's change in latitude over time isn't linear , it's a reasonable approximation for this exercise.  The spot moves about 0.18 degrees north or south in a day, all the while circling the Earth. Around the Solstice it will move 0.09 degrees north and 0.09 south. The Sun's angular size in the sky is 0.5 degrees, so, it's reasonable to conclude that yes, the Sun is directly overhead every point of the Tropic of Cancer at some time around the summer solstice.  

(The solstice does't always happen on June 21, but depending on the tolerance, some degree of direct-overheadness will be the situation  for a few days before and after the solstice).

So now, we get to your last two tricky terms: "noon" and "June 21" , which believe it or not are in conflict for this example. 

There is a local "noon" (defined by when the Sun is the highest in the sky it can get) and a clock "noon" (defined by your time zone). Only people at lucky longitudes in their time zones get to have them coincide.  But "June 21" is defined by those very time zones.  And so the sun can be "directly overhead" at "noon" on June 21 only around those lucky meridians. Elsewhere the sun will be directly overhead at some time other than noon.