Uses of type I superconductors

There must be a reason that Type I SCs are used in SQUIDs (resolution?).

 

Whether or not a superconductor is type I or II is generally of little importance in most applications. All high-Tc superconductors are type II so if you want to work at high temperatures there is no choice.
There are only a few low-Tc superconductors that are actually used and most of them are type II. Niobium and its superconducting alloys are all type II and are used in almost all real-life applications of superconductivity (including MRI magnets) since the Tc is rather high and they can carry reasonably high currents.
Anyway; the most important low-Tc superconductor for fundamental research is aluminium but that has little to do with the fact that it is type I; the main advantage is instead that aluminium-oxide is a very good insulator and can be created by simply exposing aluminium to an oxygen atmosphere during fabrication. This means that high-quality junctions can be created which in turn means that we can fabricate multilayer structures such as Josephson junctions, SQUIDs, SETs etc using relatively straightforward methods (shadow evaporation). It is MUCH harder to fabricate good Nb structures (and even then aluminium-oxide is used as the insulator).
Aluminium also has a fairly high Tc (1.2-1.6K) meaning even a simply He-3 cryostat will often do.

Most low-Tc SQUIDs are fabricated from niobium and aluminium SQUIDs are rarely used as actual magnetometers; Al SQUIDs are usually just used as "tunable Josephson junctions" since we can control the critical current (and therefore Ej) using an external magnetic field (this is used in e.g. split Cooper-pair boxes etc).

 

I have no idea what a split Cooper-pair box is, but thanks for the clarification anyway.

Or even a He-4 cryostat! (or was that a typo?)

 

Unfortunately not. A pumped He-4 cryostat will get down to 1.3-1.4K but that is still too close to Tc for most applications (the IV curve of an Al Josephson junction will be extremely smeared out at 1.3K, and it might not be superconducting at all).
Fortunately pumped He-3 cryostats are quite cheap nowadays so most labs can afford them; they will get down to about 260-300 mK and have a hold time of about 24 hours when wired up correctly. There are even cryogen-free closed-cycle He-3 systems that use cryocoolers to liquefy the He-3; meaning you do not even need liquid He-4 to cool the system (unfortunately they are very noisy; we have one where I work and the sound drives me nuts whenever I have to spend any time in that lab).