Why don't we have a "Cooper pair" of two holes in a superconductor? The condensate of Cooper pairs is described by a complex scalar field (or the order parameter) which, when quantized can give rise (or is capable of creating) two types of quanta with charges opposite to each other. But we do not have a "Cooper pair" of two holes or antielectrons but only a sea of negatively charged Cooper pairs.
 A: Actually, you certainly can have Cooper pairs of positively charged holes instead of negatively charged electrons. (Note that these are holes in the solid-state sense). Like the comments mention, you can also have positron cooper pairs in antimatter, at least in principle, but there are no known physical cases of that.
As an explicit example, in so-called "hole-doped" high temperature cuprate superconductors, the Cooper pairs are pairs of positively charged holes, not electrons. These include the well-known superconductor used in most superconductor education kits: YBCO
There is at least one theorist that thinks hole superconductivity gives higher transition temperatures than electron superconductors. See Jorge Hirsch's page here: https://jorge.physics.ucsd.edu/talks.html
A: This is a very interesting question. Yes, it is possible to have "hole superconductivity" in principle. There are actually some experimental claim of the existence of hole superconductivity in some materials. To understand what it means to be a "hole superconductor", let me clarify few points, which may led to confusion. When one talks about particles and holes there are three levels of "complexity":

*

*Electron and positron. They are the particles and holes of the Dirac field. They are charge conjugates, i.e., they are symmetric under charge symmetry $C$. These are fundamental elementary particles and $C$ is a fundamental law of physics. On this part of the universe where we live, positrons are extremely rare. Normal solids are made of electrons. So, positrons are not relevant for superconductors (on this planet).


*Conduction electrons and holes in semiconductors. At zero temperature, electrons in solids occupy all possible energy states under the Fermi level. These electrons are called valence electrons. They are basically "frozen" and cannot contribute to charge transport. At finite temperature however some of the valence electrons can jump from a lower energy level to an higher energy level. This electron will be a conduction electron because it is able to transport electrical current. The empty lower energy level is now...empty. This behaves as a "hole" and can carry electrical current. The distinction between hole and conduction electrons can be determined experimentally with the Hall conductance (which is basically the Lorentz force applied to electrons in a solid). This is possible because electrons are negatively charged, whereas holes are positively charged. Therefore the effect of the magnetic field and of the Lorentz force on electrons and holes is opposite. There is also particle-hole symmetry here, but it is not an exact symmetry and it is not a fundamental law of nature.


*Quasiparticle excitations in superconductors. Now, in a superconductor, the conduction electrons form a condensate of Cooper pairs. This is the groundstate of the system with condensation energy $E_0$. On top of this ground state, there are excitations with positive energy (respect to the groundstate), and excitations with negative energy (respect to the groundstate). These are referred sometimes as particles and holes, or more generally as "Bogoliubov quasiparticles". The ground state can be thought as the vacuum of the particles (positive-energy excitations) or the state with all hole states (negative-energy excitations) occupied. There is a particle-hole symmetry which is exact (at least for low energy excitations). The important thing is that these quasiparticles are not particles in sense of elementary particles, but are collective excitations of the condensed matter system.
Now, what I describe before in 3) are superconductors where the Cooper pairs are formed by electrons. There is also the other possibility where Cooper pairs are made by holes, where holes now means the holes that I defined at number 2. This quantum state is theoretically possible, and it is realized by electron-doped cuprates, for example, Nd$_{2-x}$Ce$_x$CuO$_4$ (NCCO).
A: Holes are a description of quasiparticles in semiconductors which have an equal or near-equal balance of electrons and holes when the material is undoped, i.e., it is an intrinsic semiconductor. At such low electron or hole densities, it is practically impossible for the semiconductor to form a superconducting system.
That being said, we do have hole superconductivity. Twisted graphene (bilayer and trilayer) exhibit superconductivity in the electron and hole-doped regime because these ultrathin 2D systems can be doped using the field effect. See https://www.nature.com/articles/s41586-021-03192-0 for example.
