I might be able to give you at least a little information about the pagan connections, but most of what I know about święconka is closely tied with the Christian tradition. Most sources point to the fact that Christianity and paganism are tightly intertwined in Easter rituals, making it more difficult to distinguish between the origin of certain aspects. The most basic answer to this I have is that the blessing of food prior to consumption during the pagan celebration of the spring equinox was equated with endowing it with the powers of healing and vitality. Unfortunately I have no sources on hand to give you, it’s something I’ve read at some point but cannot remember where. I think the blessing of the Easter basket as well as Easter breakfast, and possibility of them having pagan roots can be somewhat inferred from other connections between the pagan and Christian rituals associated with the spring season.
Prior to the advent of Christianity, Slavs celebrated Jare Święto or Jare Gody, during the time of the spring equinox. Like Easter, it was associated with resurrection, fertility, and life. The Slavic pagan god Jaryło is connected with the spring season, renewal, and fertility. Jare Święto, to my knowledge, is also the celebration of his birth. One of the most prolific symbols which unites pagan and Christian traditions in this area is the egg. Polish pisanki were decorated long before Christianity came to Poland, often with pagan symbols. Pagan Slavs attributed magic properties to them, including healing powers. Pisanki are a vital part of not only the Easter holiday itself for Christians, but of course part of the święconka basket and Easter breakfast too. The idea of rebirth and renewal is probably one of the main uniting factors between pre- and post-Christian tradition of this time of year. In the pagan tradition the decorated eggs were supposed to bring happiness, vitality, and a plentiful and healthy season of crops. (Decorating eggs, of course, reaches far beyond Slavic lands and is present in the histories of many different cultures around the world.)
Jare Święto, like Easter, included a feast. The traditional Easter basket should include bread and/or cake symbolizing life and well-being, and pagan Slavs baked kołacze (a type of circular bread) for this occasion. The ritual was also shared with the deceased, when families brought food and drink to their graves to honor and include them in the celebration, as it was believed their souls roamed the Earth during this time. (see also Dziady) Moreover, the ritual of bathing in and sprinkling one another with blessed water during this time was also observed, which is suggested to have survived in the form of Śmigus-Dyngus until today. Water, of course, symbolically and literally is a way to cleanse ourselves. Blessed water, whether in Christian or any other faith seems to have similar properties.
A blog which may be able to answer your question better, and whose author is much more knowledgeable on the topic of Polish history and culture, is lamus-dworski. Also, if any of my followers would like to bring any corrections to my answer, or perhaps have a much better insight into the topic, please add to this post.