Methods of conjugating oligonucleotides are provided. The methods may include: activating a terminal of first oligonucleotide using a squarate reagent to produce an activated first oligonucleotide; and binding the first oligonucleotide and a second oligonucleotide to a splint oligonucleotide; to conjugate the activated first oligonucleotide with a terminal of the second oligonucleotide via a squaramide linkage to produce a squaramide-linked oligonucleotide. Also provided are oligonucleotides that include a squaramide internucleoside linkage. Compositions are provided that include a first oligonucleotide including 300 or more nucleosides and at least one squaramide internucleoside linkage; and a second complementary oligonucleotide not including a squaramide linkage. Kits and compositions for practicing the subject methods are also provided.