A quarter of a long circle
A square is ninety degrees of separation, one quarter of the zodiac. With Uranus needing something like eighty four years to go all the way round, the first square lands near twenty one, the opposition near forty two, and the closing square near sixty three.
All of this is measured in ecliptic longitude, the position along the zodiac. It has nothing to do with brightness, distance or visibility, and the same arithmetic applies to every body in the chart.
The same body produces its second square near sixty three, and the pair are read as the opening and closing quarters of one long arc rather than as separate events.
Why twenty one is a range, not a birthday
Planets travel elliptical paths, so their speed changes across the orbit. Covering a quarter of the way round takes longer from the slow stretch than from the fast one, and the difference is not trivial at this scale. Depending on where Uranus sat in your chart, the first square can arrive at nineteen or hold off until twenty three.
Quoted ages are averages taken across many charts. Your own crossing comes from your natal degree, which is why the tool works from a position rather than adding a number to a birth year.
Anyone comparing notes with friends of the same age will find the dates scattered across several years. That scatter is the orbit and not a disagreement between ephemerides, which agree with each other to within minutes.
There is a second reason to compute rather than assume. People often take the square as landing on or near a birthday, which it rarely does. The transit is set by a planetary position, and the crossing can fall in any month of the year with no reference to the date you were born.
The retrograde question at this age
Uranus spends about five months of each year in retrograde motion. When a square falls close to one of the stations, the aspect is exact three times: once going forward, once going back and once going forward again. Several months to a year can separate the first crossing from the last.
Where the timing avoids the retrograde altogether there is a single crossing and the whole thing is over quickly. Astrologers read the full span as the transit either way, and the calculator above shows which case applies to you.
Whether you get one crossing or three is decided entirely by the calendar, since it depends on how the transit lines up with the retrograde period of that particular year. There is no interpretive weight in the difference beyond the length of the span.
How it is read, and the part that needs a birth time
The convention reads this square as the first serious test of the Uranus theme in a chart, arriving at an age when most people are leaving education or a family home. Independence, impatience and a wish to do things differently are the usual words. The tradition also allows for the awkward side of that, since decisions made from restlessness at twenty one are not always ones that hold. It is interpretation, marked as interpretation, and no event is being predicted here.
The house of your natal Uranus is what narrows the reading, and it requires a recorded birth time. The dates do not, since Uranus barely moves in a day.
Age is doing a good deal of the work in the usual descriptions of this transit, and that is worth keeping in view. Much of what is attributed to Uranus at twenty one would be said of twenty one in any case.
Questions people ask about this
When does Uranus square Uranus happen?
Around age twenty one, with a genuine spread from about nineteen to twenty three. Where your natal Uranus sat in the orbit decides which end of that you fall on.
Why is the age not fixed?
Because the orbit is an ellipse and Uranus does not keep a constant speed. A quarter of the circuit takes noticeably longer when it starts from the slower part of the path.
Can it happen three times?
Yes, if the crossing falls near a retrograde station. Uranus then reaches the exact angle going forward, again in reverse and a third time going forward, over a span of months.
Is this the same as the Saturn return?
No. The Saturn return is Saturn arriving back at its own natal degree near twenty nine, while this is Uranus at a right angle to its own at about twenty one. Different bodies, different geometry.
What happens at the later Uranus squares?
The closing square arrives around sixty three, a quarter of a lap before the cycle completes near eighty four. The opposition sits between them in the early forties.
Do I need my birth time?
Not to get the dates. You need it for the house placement, which is the difference between a general description and something specific to your chart.