Do wireless router boosters really work, and if so what is recommended?
Is it better to upgrade to a high gain antenna instead of a signal booster?
My router is in a upstairs bedroom, on an outside wall.(House is 2100 sq ft.)

Wireless booster only work if they are two way, most are one way so dont work.

They are often illegal as well..

The antenna thing is a whole different ball game,

They can boost the range (send) but if the reciver is not powerful enought to return the signal then there wont be a connction.

BUT if you have a antenna at the base, which has a more powerful reciver, then it may work.

Remember its illegal to boost the power up, but its not illegal to boost the power down, so you can reciver 3000000DB but you cant send over 6DB

Boosting the recive is as good as boosting the send.

Actually, any general consumer modification to RF products in certain countries is illegal and theoretically can result in confiscation of equipment and fines. That includes attaching any antenna not specifically approved for that specific product.

As a practical matter however, that's unlikely to happen.

And, just because someone sells an antenna and says you can attach it to something, doesn't mean it's legal to attach it to that something. It's up to the user to make that determination.

That being said, the antenna on both ends generally play the most important role in any radio communication system, for better or worse. In some cases the wire leading to a remote antenna is so lossy, or is the wrong impedance for a given frequency, that any possible benefit from the antenna itself is eliminated.

There are also certain licenses one can obtain (e.g. Amateur Radio Operator at various levels) that permit one to make modifications or operate radio equipment under certain conditions unavailable to the general consumer.