If you were place a strong permanent magnet into the solar wind, like a 1 inch cube neodymium magnet, would the charged particles of the solar wind colliding with the magnet's magnetic field push the magnet out of our solar system, that being if it does not encounter any planets or objects in its path?
No. The magnetic field has two poles, the force on the two pole of the magnet is equal and opposite. This is why a compass needle will align in the North South direction, but is not pulled towards the North or the South.
Two magnets will attract or each other because the North pole of one is nearer the South pole of the other. You can get a net force on a magnet when the field-lines are getting bunched together. But the magnetic field in the plasma is not like this. To a first approximation they are parallel to each other and perpendicular to the equator of the sun. In fact the magnetic field is complex as it is carried with the flow of solar wind.
There would be no net force on the magnet, but it may tend to align with the magnetic field. This might cause a turning force, but no acceleration.
There would be some force due to solar wind particles impinging on the object, but since there is only something like 5 atoms per cm3, the force isn't very big, and it has nothing to do with the object being a magnet.
If you were place a strong permanent magnet into the solar wind, like a 1 inch cube neodymium magnet, would the charged particles of the solar wind colliding with the magnet's magnetic field push the magnet out of our solar system ...
Technically speaking, yes. Practically speaking, it would take an extremely long time. The solar wind consists of 5 atoms per cubic centimeter per second. Solar sails are usually huge and work by reflection, simply pushing on a tiny object is inefficient, being magnetic helps very little.
The Sun's gravity would pull in any large object not already in orbit, that would be dominant over magnetic forces from the very weak wind. If the object was already in orbit, the situation would be akin to dust particles leaving a comet-- their path is more affected by light pressure than magnetic pressure, but for a large object, neither of those could rival gravity.