Quantum states are highly unstable and subject to noise, so with current levels of technology it is unlikely that this entanglement would last for any useful amount of time.
But let's assume that you have two entangled fault-tolerant qubits (one owned by a malicious manufacturer), and that they are in superposition (in the Z-basis):
$$
\frac{1}{\sqrt{2}} (|01\rangle + |10\rangle)
$$
Where the first qubit is owned by the manufacturer. These are perfectly anti-correlated, meaning that a measurement of 0 on the first qubit causes the second to be measured as 1.
This may seem promising if you are a nefarious manufacturer; however the moment the user mesures their qubit, they collapse the superposition and the two qubits are no longer entangled, you have two separated states $|0\rangle$ in the user's computer and $|1\rangle$ in the manufacturers. You cannot bring them back into superposition without some interaction between the two qubits again.
Quantum computations are repeated many times to get the expected value. With this scheme you get only one of the (probably millions of) repeats.
And as @Sanchayan Dutta mentions, the choice of basis is important (this is what we communicate classically in teleportation), if the user were to rotate their measurement basis randomly before performing the measurement, the manufacturer would be unaware of this. The manufacturer's qubit would be in an eigenstate of the basis, but if they then chose to measure using the Z basis, it would collapse into an eigenstate of that basis.