There are several sources of confusion resulting from misunderstood and loosely used words.
Potential energy is different form potential.
Potential energy (symbol U, unit Joule) is a general concept, seen in mechanics, as well as in electricity. In words, the force on a particle, whether it is a point mass in a gravitational field, or a point charge in an electric field, is in the direction of decreasing potential energy. It does not matter whether the charge is positive or negative. In both cases, the force on the charge is in the direction of decreasing potential energy. Certainly, it is true that absolute value of potential energy U is not defined, only differences in U are observed. But once the zero of U is defined at a particular point, it is defined everywhere.
Electric potential, or just potential (symbol V, unit Volt) is defined like this: At every point in space, there is a scalar quantity V, such that, if you place a point charge q (positive or negative) at that point, the potential energy is (q)(V). This is so whether the charge q is a positive or negative number. Similar to potential energy, absolute value of V is not defined, only differences in potential are observed.
If you take a 1.5 V battery, and you define the negative terminal to be 0V, the positive terminal is 1.5V. Now, if you place a +1C charge at the positive terminal, the potential energy is (+1C) (1.5V) = 1.5 J. If you place the positive charge at the negative terminal, the potential energy is (+1C) (0V) = 0J. If you place a - 1C charge at the positive terminal, the potential energy is (- 1 C) (1.5V) = - 1.5 J. If you place it at the negative terminal, the potential energy is (- 1 C) (0 V) = 0 J.
You can redefine the zero of potential, and can say, for example, that the positive terminal of the battery is at 0 V. Then automatically, the negative terminal is at - 1.5 V. The battery makes sure that it maintains the potential difference. The method of calculation of potential energy is the same, use U = qV, and recalculate all the values again.
In every case, you will notice that the positive 1C charge has a high potential energy at the positive terminal and low potential energy at the negative terminal. The negative charge has a high potential energy at the NEGATIVE terminal, and a low potential energy at the POSITIVE terminal.
Now use the rule that everything (mass, charge, does not matter) is pushed towards lower potential energy, and you will see that negative charges flow from the negative terminal of the battery, along the wire, towards the positive terminal, and positive charges flow from the positive terminal to the negative terminal