Hydrogen is liquid below 21.15 K and goes solid at 14.01 K.
Helium is liquid below 4.15 K and goes solid at 0.95 K under huge pressure of 2.5 MPa.
So it should be possible to use gaseous helium to sub cool liquid hydrogen. It is gaseous even below the temperature of solid Hydrogen.
Methane boils at 111.65 K and solidifies at 90.7 K. Sub cooling it with gaseous helium should be possible too. According to this NASA paper the solubility of helium in subcooled liquid methane is small, about 0.0001 mole fraction at 25° F (13.9 K), but these temperature values seems to be wrong. In a graph a value of 98 K is found on page 20.
Helium does not react with hydrogen or methane. I don't know if helium is soluble in liquid hydrogen or methane. But helium is still gaseous in a liquid air separation plant without using an extra cooling stage for Helium. Helium is used to pressurize liquid hydrogen in rockets, so its solvability should be at least small if not zero.
According to this page there might be a hot metallic alloy of hydrogen and helium within the core of Jupiter and Saturn. But this alloy exists only under enormous pressure.
Here is some information about solubility:
The solubility of helium in liquid hydrogen was measured as a function
of temperature and pressure. It was found that the solubility varied
from 0.59 mole% for T = 26.8°K and 1.98 atm partial pressure of helium
to 11.1 mole% for T = 19.8°K and 7.0 atm partial pressure of helium.
But the partial pressure of helium in a LH2 rocket tank would be lower than 7 bar.