As one can deduce from the information provided in A Forgotten Mathematician, he was Dutch, confirming the Wikipedia claim.
According to 1,
Egbertus Rudolf van Kampen, known as Egbert, was born on 28 May 1908
in Berchem as the youngest in a family of three children. Egbert’s parents had moved from the Netherlands to Belgium a couple of years before, when his
father became an accountant at the Minerva car factory in Antwerp.
The history of the Belgian law on nationality is discussed in great detail in
Marie-Claire Foblets, Zeynep Yanasmayan,
Patrick Wautelet, EUDO Citizenship Observatory: Country Report: Belgium.
To summarize, until 1909, Belgian citizenship rules were modeled on the those in France:
(1) Citizenship was granted at birth, following the principle of ius
sanguinis paterni: a child whose father was French had French citizenship, even if she or he was born abroad (art. 10 Code Civil). This mode of acquisition was legitimised by the nationalist conviction that citizenship cannot simply follow from accidentally having been born in a certain territory, but has to be seen as the heritage of a people, which is made up of individuals who together form a sovereign nation (Verwilghen 1985: 18; Closset 1984: 782).
(2) If a foreigner was born in France, he or she could nevertheless voluntarily opt for French citizenship by making a déclaration de domiciliation [declaration of domicile] within one year after reaching the age of 22 (art. 9 Code Civil). The theory of Jean-Jacques Rousseau was put into practice by means of this policy. According to Rousseau the acquisition of citizenship constitutes a contract to which an individual can voluntarily choose to adhere.
(3) The principle of ‘one family, one citizenship’ was also stressed (arts. 12 and 19 Code Civil). Until 1985, the Belgian legislation continued to be based largely on this principle: a married woman took the citizenship of her husband and so did the children born in wedlock. The loss and re-acquisition of French citizenship were regulated in arts. 17 and 20 of the Code Civil.
The law of 1909 thus abrogated the provisions of the Code Civil regarding citizenship
and of certain specific laws,10 and introduced a whole new set of rules. This law was liberal for the time of its promulgation, as it provided for the application of the ius soli principle beside ius sanguinis. More precisely, citizenship was also granted to every child born in Belgium of parents with an undefined status and to all persons turning 23, who had lived in Belgium during their 22nd year and did not indicate a desire to retain their foreign citizenship, provided that either the person was born in Belgium and had been domiciled in Belgium for at least 6 years or the person was born in Belgium of foreign parents, at least one of whom was also born in Belgium or had resided there continuously for ten years.
Thus, initially, Egbert Van Kampen did not qualify for Belgian citizenship. But he did not qualify after the laws of 1909 either, since his family moved back to Holland during WWWI (1914-1918: Holland was neutral) and, subsequently, Egbert Van Kampen went to Gottingen to study, then to Hamburg to work and, eventually, moved to the United States (1931), to Johns Hopkins University.
To summarize: Van Kampen could not have been a Belgian citizen.
Turning to the Dutch citizenship laws: The history is long and complex (since the country is much older than Belgium), see this Wikipedia article:
Nevertheless, the new 1850 nationality law did not replace the 1838 civil code and this in turn led to the creation of a double nationality; political nationality as stated in 1850, and civil nationality of the 1838 civil code. The uncertainties were eliminated in 1892 with the implementation of the Nationality Act which replaced both 1838 and 1850 nationality acts. The new law was based on the German system of jus sanguinis and excluded all previous mention of the principle of jus soli. The new Dutch nationality law survived until the amendments of 1985.
Thus, (in view of the jus sanguinis system) Van Kampen, as a child of Dutch parents (both his parents, Laurens van Kampen and Johanna Maria Schiltmeijer were born in Holland, see here), was automatically Dutch by birth.
You can find detailed information on Van Kampen's life and work in the article
1 Robbert Fokkink, A forgotten mathematician, Newsletter of the European Mathematical Society, 2004, Issue 52, pp. 9-13.
For his article (details on Van Kampen's early life) Fokkink interviewed Van Kampen's relatives as well as local officials: Mark de Bock from Antwerp and Albert Schiltmeijer from Amsterdam.
Lastly: Van Kampen lived in the US for about 10 years (1931-1942, he died very young), so, according to the US naturalization laws of the period, he would have qualified for the US citizenship as early as 1936. I do not know if he ever became a US citizen.