Electromagnetic interactions, in a sense, are responsible for all motion we see on a daily basis, including picking up and moving objects, but it definitely isn't as cool as telekinesis. I'll first correct some of the physics in the question and then try to provide some solutions which don't break existing physics.
First some notes on the laws which must be followed:
Conservation of mass: Conservation of mass is only approximate for most commonly encountered terrestrial situations. Nuclear interactions for instance often do not conserve mass. They do conserve mass-energy however, so this law may be combined with the conservation of energy.
Square-cube law: is a mathematical law and only valid for Euclidean geometry, since we really won't encounter situations that are non-euclidean on a daily basis, it is a safe assumption.
Now on to the physics:
"What I'm thinking is that they can either manipulate gravity or electromagnetic fields to some degree to strengthen or weaken the bonds between electrons so they can either attract or repel."
Electrons do not bond, bonding involves a negative energy potential which ensures that energy must typically be added before the bond can broken. Bonds place restrictions on the motion of the particles bound, and ensure that their movement is localized.
Electrons only interact through the electro-weak interaction. At low energies (those in which electrons will remain bound to an atomic nucleus) they mainly interact through electromagnetic interaction (the weak nuclear interaction is only relevant in nuclear decay processes). Since all electrons are negatively charged, the potential of the interaction between them will be strictly positive and there will be a force which pushes two or more electrons apart.
If they were able to manipulate electromagnetic fields, this means they would have to rearrange the electrons within the atoms of the objects they want to manipulate so that they are all spinning in the same direction. It's possible they could also magnetize non-magnetic materials through light.
Electrons are arranged by orbitals, and the arrangement is determined by quantum phenomena known as spin. The spin interactions must obey a set of rules. Electrons are spin-1/2 particles and as such as they obey the Pauli exclusion principle which states that identical particles with the same spin cannot occupy the same orbital which is why only two electrons are allowed together in the most basic energy orbital called the S orbital.
It is important to note that the spin interaction is a strictly quantum effect and is not dependent on the electromagnetic interaction. For instance, neutrons and protons interact via the strong force, but they also obey spin-1/2. This means that manipulating matter via the electromagnetic interaction is not going to allow for spin alignment of all electrons in an atom, which is forbidden by quantum laws.
First linked article
The article describes how aligned magnetic fields can be induced in certain non-ferrous metallic materials with illumination from special frequencies of laser light. I do not really see a way of harnessing this phenomena in a biological system. Lasers can be created from from biological systems:
But the power of such systems is very very weak and the energy required fairly large. For reference, the effect described in Science Daily article required about a power transmission of 30 W/cm at 25 THz for the effect to be reasonably significant. For reference, the total power output by a average human is around 100 W for normal daily activity.
Now the elephant in the room is exactly how they'd be able to channel all this...The question is where would it be located and what size would it have to be?
The linked article in this section discusses using nano-diodes to collect energy from background infrared radiation in the atmosphere. I didn't see any information about the potentials generated from these interactions and I'm guessing they are very low. It is an interesting concept however, and could (with some sci-fi handwaving) be incorporated into your story.
Now for the hard part...Telekinesis is not an easy thing to explain in an even remotely realistic way. That said, there are some directions which the science could take which would sound more realistic than others.
Electromagnetic fields are not really feasible since they require maintaining high electric potentials (strong enough to move everyday objects) and exercising great control over these electric potentials, neither of which is compatible with biologic systems.
Gravity manipulation is a possibility since gravity is not understood. Current theory explains gravity as a geometric construct and the standard model has not been able to incorporate gravity beyond mass (There is the discovery of the Higgs Boson, but this theory is far from complete, and some argue there are now more questions than answers). So using some sci-fi rigmarole by which "gravitons" are manipulated through the superheros mind could be used without explicitly breaking any conditions ruled out by preexisted physics.
Excluding cheat methods of telekinesis (using nanobots..etc.) another not-explicitly-physics-breaking explanation of telekinesis would be a fifth fundamental force. This force may be explained as being induced by an improbable set of circumstances taking place in the superheros body. This way, you could make the particles produced by this force interact with the electromagnetic interaction and/or masses over a specific range and with specific rules which allow for manipulation of matter in a manner consistence with telekinesis for all intents and purposes.