I would like to get an understanding of potential dangers.
Any time you push wood into that spinning blade, the part between the blade and the fence is at risk of binding and being thrown across the room in an event called "kickback".
This can happen with any tool with a spinning blade, but the table saw is the most dangerous because the wood becomes a missile when this happens. With a miter saw, for example, the kickback shoves the wood into the corner between the fence and the base, causing the motor/blade/handle assembly to rise back up. With the miter saw, the blade is automatically protected by the closing blade guard, and the whole thing is mechanically limited in its travel distance.
It can also happen when cutting with a hand-held circular saw. Usually, though, the saw will stop (especially if it's battery powered) because the blade will bind in the wood and it won't launch too much wood/equipment too far.
To give you an idea of the forces involved in table saw kickback, there was a 2x4 shaped chunk taken out of the chalk board* in my Junior High School shop class (back in the dark ages when there was such a thing as "shop class"). This was caused by someone attempting to rip cut a 2x4 on the table saw. The 2x4 got caught and kicked back, throwing the board across the room and hitting the black board. Note that this board flew at least** 30 feet (~10m) and traveled above all the student desks and the teacher's desk before hitting the chalk board. Fortunately, everyone was in the shop area and not in the desk area, so nobody (but the saw operator) was in the line of fire.
When cutting small pieces such as your 1x1" squares, you should always use a push stick on the piece between the blade and the fence. The stick protects your fingers from the blade; it also ensures even pressure on the wood as it travels, reducing the risk of twist which increases the risk of kick back; and it ensures that you can push the off-cut out from between the blade and fence as leaving it there also increases the risk of that piece being thrown back.
*As a follow up, that chalk board wasn't replaced during the 3 years I attended that school and probably longer. Partially because replacing it wasn't cheap, but also because it was used as an object lesson in the dangers of kickback when using the table saw. It was quite sobering to a group of early-teenage kids walking into class and seeing that dent that wasn't there yesterday. Turns out, it had happened during the class right before mine.
**Junior High School was ahem a few years ago. It was probably more like 50 feet, accounting for the rows of desks, but my memory isn't precise enough to recall exactly. I'd rather underestimate than be accused of fearmongering and over exaggeration.