The only non-religious private school that I'm aware of it Oak Hill Academy - it's very expensive. There are Catholic Alternatives (O'Hara & Marist) and then there are some Christian alternatives (Eugene Christian School, I think?)
The reason I was so excited about 4J schools was the choice - there is one Montessori charter that is now in it's fifth year - Ridgeline. I don't know much about it to be honest, but it's an option. The Village School is also a Charter school. Again - don't know much about it. Additionally, there are non-charter but alternate format schools - Charlemagne (French Immersion), Buena Vista (Spanish Immersion), a Japanese Immersion and some Arts/Technical formats. There used to be more, but they've closed several in the last 4 years. The Alternative schools tend to be at risk for school closure because Eugene is on a path of reinforcing neighborhood schools and adjusting to declining enrollment, so I would be wary of alternative schools, I think. You can find the school descriptions for both Charter and Alternative (as well as neighborhood) on the 4J website.
We are in a neighborhood school. It's been ok, but it will be really different next year. It's hard to imagine how the teachers will deal with the class sizes. My complaint with our school is that there is not enough differentiated learning options available, poor teachers are allowed to continue, and it's easy even now for kids to get lost in the shuffle. I know what I just wrote could be applied to any public school. We are lucky to be in a school that has successfully fundraised to cover gaps in music and PE, but there is some inequity among schools because of that fundraising discrepancy. Also - we lotteried into this school because of choice, and every year they talk about closing a school close by, they tell us we could lose our spot if they do so. At best, they wouldn't be able to admit my youngest, so I'd have kids at different schools. I mention that just to say - take care in the school you choose. Probably best to live in the neighborhood (of course that is ideal really from almost any perspective). Since we started in school, choice in middle and high school has also been severely restricted, so we will be forced to move in order for my daughters to track with their friends through middle school and high school. That is actually one of the big reasons we are moving out of state - if we have to move anyway, do we really want to stay here? (It's also work related - there just aren't enough jobs in Eugene).
I'll PM you the name of our school.
Good luck with your move! I don't want to make it sound like I'm totally down on Eugene, I'm not - just haven't been too impressed with the education.