You have exact equations for the solution in the related question Time it takes for temperature change. Here I would add a few comments.
It is actually easier if container is thick! Then suppose that all water is at same temperature $T$ and all the air in the frizer is at the same temperature $T_e$. $T_e$ is constant.
If that is so, you can use only Fourier's law to describe how heat $Q$ leaves the container
$$\frac{\text{d}Q}{\text{d}t} = \frac{\lambda A}{d} (T-T_e).$$
$d$ is thickness, $A$ area and $\lambda$ thermal conductivity of the container.
Knowing that water cools as heat is leaving the container
$$\text{d}Q = m c \text{d}T,$$
where $m$ is mass and $c$ is specific heat capacity of the liquid, you get rather simple differential equation
$$m c \frac{\text{d}T}{\text{d}t} = \frac{\lambda A}{d} (T-T_e),$$
$$\frac{\text{d}T}{(T-T_e)} = \frac{\lambda A}{d m c } \text{d}t = K \text{d}t,$$
which has exponential solution:
$$K t = \ln \left(\frac{T-T_e}{T_0-T_e}\right),$$
$$T = T_e + (T_0-T_e) e^{-Kt}.$$
Hmm, you're right. High-quality phase diagrams don't seem to float around for free on the web. I suppose I shouldn't be as surprised by this as I am.
Reading from this random page it looks like at 6 atm the boiling point for nitrogen is around 85 K. Oxygen is a little harder to find, since there has been news in the past few years about metallic and superconducting oxygen at very high pressures. It looks like you can buy the data to generate your own phase diagrams from Wolfram|Alpha for a few dollars; squinting at the tiny diagram that's available for free, I guess that the boiling point at 0.6 MPa is around 100–110 K.
Probably the behavior of a nitrogen-oxygen mixture (assuming that water and CO2 would freeze out) would be comparable to their behavior at one atmosphere, as shown here: a boiling that varies roughly linearly from that of nitrogen to that of oxygen as the oxygen concentration changes.
This may be more of a cryogenic engineering question.
Best Answer
The critical temperature of oxygen is -181.5 deg F. If your freezer can't produce a temperature somewhat below this, it will be impossible to liquify oxygen, regardless of how much pressure is involved.