It appears that there is little scientific backing for Jung's theories. As these theories were first suggested at the start of the 20th century they have had a considerable amount of time to receive scientific support. If they were accepted by scientists it therefore seems like they would have been widely used and cited by scholars in the interim period.
However, if I do a search (in titles, abstracts and keywords) for related content on SCOPUS I get only 12 references (the details of which I have pasted below for you to consider). The low level of search results (and the fact that few of these appear relevant) provides strong evidence that there is little scientific support for Jung's work.
In terms of alternative theories to consider, if I were you I would have a look at the "big five" also known as OCEAN as there is a lot more support for this personality theory. For examples of some influential and interesting articles, please see the OCEAN references section below.
References from the SCOPUS literature search:
TITLE-ABS-KEY ( jung "cognitive functions" )
Novak M. Ideal types of law from the perspective of psychological typology [Idealni tipi prava v lu?i psihološke tipologije] 2013 Revus
Lo J.-J., Chan Y.-C. Design of adaptive web interfaces with respect to student cognitive styles 2011 Advances in Intelligent and Soft Computing
Semetsky I. Jung and Tarot: A Theory-practice Nexus in Education and Counselling 2012 Jung and Educational Theory
Connolly A.M. Cognitive aesthetics of alchemical imagery 2013 Journal of Analytical Psychology
Naghavi H.R., Nyberg L. Integrative action in the fronto-parietal network: A cure for a scattered mind 2007 Behavioral and Brain Sciences
Sebastian A., Jung P., Krause-Utz A., Lieb K., Schmahl C., Tüscher O. Frontal dysfunctions of impulse control - A systematic review in borderline personality disorder and attention-deficit/hyperactivity disorder 2014 Frontiers in Human Neuroscience
Segall J.M., Allen E.A., Jung R.E., Erhardt E.B., Arja S.K., Kiehl K., Calhoun V.D. Correspondence between structure and function in the human brain at rest 2012 Frontiers in Neuroinformatics
Vieira A.G., Sperb T.M. Symbolic play and the narrative organization of the child's life experience [O brincar simbólico e a organização narrativa da experiência de vida na criança] 2007 Psicologia: Reflexao e Critica
Park H.-J., Lee S.Y., Shim H.S., Kim J.S., Kim K.S., Shim I. Chronic treatment with squid phosphatidylserine activates glucose uptake and ameliorates TMT-induced cognitive deficit in rats via activation of cholinergic systems 2012 Evidence-based Complementary and Alternative Medicine
Seo Y., Burns K., Fennell C., Kim J.-H., Gunstad J., Glickman E., McDaniel J. The Influence of Exercise on Cognitive Performance in Normobaric Hypoxia 2015 High Altitude Medicine and Biology
Kwon H.G., Choi B.Y., Kim S.H., Chang C.H., Jung Y.J., Lee H.D., Jang S.H. Injury of the cingulum in patients with putaminal hemorrhage: A diffusion tensor tractography study 2014 Frontiers in Human Neuroscience
Lee J.H., Kim S.E., Park C.-H., Yoo J.H., Lee H.W. Gray and White Matter Volumes and Cognitive Dysfunction in Drug-Naïve Newly Diagnosed Pediatric Epilepsy 2015 BioMed Research International
OCEAN sample references
Roberts, Brent W., Kate E. Walton, and Wolfgang Viechtbauer. "Patterns of mean-level change in personality traits across the life course: a meta-analysis of longitudinal studies." Psychological bulletin 132.1 (2006): 1.
Hayes, Natalie, and Stephen Joseph. "Big 5 correlates of three measures of subjective well-being." Personality and Individual differences 34.4 (2003): 723-727.
Kotov, Roman, et al. "Linking “big” personality traits to anxiety, depressive, and substance use disorders: a meta-analysis." Psychological bulletin 136.5 (2010): 768.