There is a nice answer by annav, I would like to add a few interesting notes.

Most of the answers display the proton and the neutron as consisting only of three quarks, in reality, it is much more complex.

https://profmattstrassler.com/articles-and-posts/largehadroncolliderfaq/whats-a-proton-anyway/

Protons and neutrons consist of a sea of quarks, antiquarks and gluons. I admit it is tempting and easy to use the valence quarks only.

You are suggesting a combination (by rearranging the quarks, the carriers of the EM force between the nucleons) of valence quarks where the total (net) EM charge of these new hadrons would have to be half of the elementary (the electron's) charge, but we happen to live in a universe where to our best knowledge (even though there is pion exchange) this cannot happen because:

1. mathematically, you cannot create a combination of quarks, that would yield you the required half elementary charge, that just cannot happen. So you have to give up on that, but your question could still be interpreted as "can we rearrange the quarks so that the nucleons have equal EM charges?". Now if you only allow three valence quarks (up and down like in the case of the proton and neutron), knowing the EM charges of the up and down quarks, there is no combination that would yield equal EM charges for the nucleons.

Exotic hadrons are subatomic particles composed of quarks and gluons, but which - unlike "well-known" hadrons such as protons , neutrons and mesons - consist of more than three valence quarks.

https://en.wikipedia.org/wiki/Exotic_hadron

2. if you allow more (or less) then three valence quarks, then theoretically you could just have all kinds of exotic mesons, or tetraquarks, pentaquarks, hexaquarks (or other such combinations), until you get equal EM charge for both nucleons. The problem is stability, because to our knowledge, this (or any other such) combination is not stable (can't create a stable nucleus, not to mention the interactions with the electron field). Quarks live in confinement, and there has to be a balance between the forces to make a hadron stable (please note the even the neutron is only stable inside the nucleus).

Again in rough lines, they are quantum mechanically bound by the potential with the additions of two fundamental forces, the strong QCD force, and the electromagnetic force. In quantum mechanics, as stated also in the comments, systems are stable when they are in the lowest energy state, and this state happens to be the proton.

Why is the proton the only stable hadron?

So the ultimate answer to your question is balance between the forces (strong, EM, and PEP), some kind of symbiosis between these forces that in our universe just happens in case of the proton and the neutron, but these do not satisfy your wish for equalizing the EM charge. Bottom line, to our knowledge, there is no combination of quarks, that would satisfy both stability (this balance between the forces) and your wish for equalizing the EM charge.

There is a nice answer by @annav, I would like to add a few interesting notes.

Most of the answers display the proton and the neutron as consisting only of three quarks, in reality, it is much more complex.

https://profmattstrassler.com/articles-and-posts/largehadroncolliderfaq/whats-a-proton-anyway/

Protons and neutrons consist of a sea of quarks, antiquarks and gluons. I admit it is tempting and easy to use the valence quarks only.

You are suggesting a combination (by rearranging the quarks, the carriers of the EM force between the nucleons) of valence quarks where the total (net) EM charge of these new hadrons would have to be half of the elementary (the electron's) charge, but we happen to live in a universe where to our best knowledge (even though there is pion exchange) this cannot happen because:

1. mathematically, you cannot create a combination of quarks, that would yield you the required half elementary charge, that just cannot happen. So you have to give up on that, but your question could still be interpreted as "can we rearrange the quarks so that the nucleons have equal EM charges?". Now if you only allow three valence quarks (up and down like in the case of the proton and neutron), knowing the EM charges of the up and down quarks, there is no combination that would yield equal EM charges for the nucleons.

Exotic hadrons are subatomic particles composed of quarks and gluons, but which - unlike "well-known" hadrons such as protons , neutrons and mesons - consist of more than three valence quarks.

https://en.wikipedia.org/wiki/Exotic_hadron

2. if you allow more (or less) then three valence quarks, then theoretically you could just have all kinds of exotic mesons, or tetraquarks, pentaquarks, hexaquarks (or other such combinations), until you get equal EM charge for both nucleons. The problem is stability, because to our knowledge, this (or any other such) combination is not stable (can't create a stable nucleus, not to mention the interactions with the electron field). Quarks live in confinement, and there has to be a balance between the forces to make a hadron stable (please note the even the neutron is only stable inside the nucleus).

Again in rough lines, they are quantum mechanically bound by the potential with the additions of two fundamental forces, the strong QCD force, and the electromagnetic force. In quantum mechanics, as stated also in the comments, systems are stable when they are in the lowest energy state, and this state happens to be the proton.

Why is the proton the only stable hadron?

So the ultimate answer to your question is balance between the forces (strong, EM, and PEP), some kind of symbiosis between these forces that in our universe just happens in case of the proton and the neutron, but these do not satisfy your wish for equalizing the EM charge. Bottom line, to our knowledge, there is no combination of quarks, that would satisfy both stability (this balance between the forces) and your wish for equalizing the EM charge.