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The only place where a spider could lay eggs inside the human body is in the ear canal (or possibly in the nasal passage... but this would be quite rare). Spiders do not lay eggs under human skin but there are some species that can lay eggs under caterpillar "skin".
A spider can not lay eggs under human skin so this lesion is definitely not spider-related. On the other hand, there are common bacterial infections that can involve the follicles of the nose-hairs. In most cases these can be treated by topical over-the counter antibiotics but if the lesion is large or growing, oral antibiotics are necessary.