I don't think you have anything to worry about. I've never heard of anyone planting corn and potatoes together, and corn and squash together shades out the squash while not benefitting the corn.

About the only real no-nos are onions/garlic and peas/beans. The onion roots produce an allelopathic chemical that inhibits growth of legumes so they have to be a couple of feet apart so that the roots don't interact. And you shouldn't plant peas/beans where onions were grown the previous year or you will have reduced yield.

Other companion planting ideas are based on common sense grouping cool season with cool season, warm with warm, and don't plant something you have to dig up with something that stays in the ground all season. Plant tall behind short etc.

Trap crops like Radish and Collards that insects prefer over the companion should be one row over. Aphid magnates like nausturtiums just have to be close by so that the predatory insects after the aphids can fly over to the crops. Same goes for most flowers.

A big misconception has to do with planting legumes (beans and peas) with other crops. If there is enough nitrogen in the soil for the other crop to grow well the legume will use it and not fix any for themselves. Beans are naturally poor nitrogen fixers anyway, and those crops that do fix a significant amount of nitrogen only benefit the soil the next year after they have been plowed under and decompose.

Alot of the companion planting stuff is non scientific and based on the color intensity of extracts from paired plants.

http://www.gardening.cornell.edu/factsheets/ecogardening/complant.html

According to Iroquois legend, corn, beans, and squash are three inseparable sisters who only grow and thrive together. This tradition of interplanting corn, beans and squash in the same mounds, widespread among Native American farming societies, is a sophisticated, sustainable system that provided long-term soil fertility and a healthy diet to generations. Growing a Three Sisters garden is a wonderful way to feel more connected to the history of this land, regardless of our ancestry.

Here is a link that might be useful: companion planting chart

Spiced Ham,
I mistakenly planted Super Sugar Snap peas on a pea/bean tower in the corner of a 3'x3' growbed. (my mistookie!) The remaining 3/4 of the bed is planted with 15 GARLIC plants. Unfortunately, I revisited my gardening books and read about INcompatible plants after planting the peas. There's a slight slope, with the peas on the downside of the bed. Everyone's happy so far, but I'll be curious to see how things turn out esp. for the peas.

I'm not positive about the strength of the garlic, but it is a close relative to onions (both are alliums) so I assume that it will have an affect (as would shallots). You might want to do a search for posts about onions and peas to see how close together peopl see an effect. Steve Solomon's book on organic composting (free online) shows a diagram of his beens doing poorly in a bed where they partially overlapped with the previous year's onions.